An Exact Formulation of the Time-Ordered Exponential using Path-Sums

P.-L Giscard (1) , K. Lui , S. J. Thwaite , D. Jaksch

1 ULCO - Université du Littoral Côte d'Opale

We present the path-sum formulation for the time-ordered exponential of a time-dependent matrix. The path-sum formulation gives the time-ordered exponential as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as selfavoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on a graphs and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the time-ordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.

Domaines: Analyse classique [math.CA]; Combinatoire [math.CO]; Systèmes dynamiques [math.DS]; Physique mathématique [math-ph]

P.-L Giscard, K. Lui, S. J. Thwaite, D. Jaksch. An Exact Formulation of the Time-Ordered Exponential using Path-Sums. 2022. ⟨hal-03606308⟩